Stéphane Dellacherie , Gloria Faccanoni , Bérénice Grec ... · j x x 0j2 1 fj x 0j

ESAIM: PROCEEDINGS AND SURVEYS, September 2014, Vol. 45, p. 138-147J.-S. Dhersin, Editor

2D NUMERICAL SIMULATIONOF A LOW MACH NUCLEAR CORE MODELWITH STIFFENED GAS USING FREEFEM++

Stéphane Dellacherie1, Gloria Faccanoni2, Bérénice Grec3, Ethem Nayir3

and Yohan Penel4

Abstract. We investigate a simplified model describing the evolution of the coolant within a nuclearreactor core (e.g. of PWR type). This model is named Lmnc (for Low Mach Nuclear Core) and consistsof the coupling between three equations of different types together with boundary conditions specificto the nuclear framework. After several articles dedicated to dimension 1, we present in this papersome monophasic two-dimensional numerical results when the fluid is modelled by the stiffened gas lawdescribing the pure liquid phase. The underlying numerical strategy is based on the Finite-Elementsoftware FreeFem++.

Résumé. Nous étudions dans ce document un système d’équations modélisant le comportement sim-plifié du fluide caloporteur dans un cœur de réacteur nucléaire (par exemple de type REP). Ce modèlea pour nom Lmnc (pour Low Mach Nuclear Core) et se compose de trois équations de natures dif-férentes couplées avec des conditions aux limites spécifiques au domaine nucléaire. Après plusieursarticles dédiés à l’analyse du modèle en dimension 1, nous présentons ici des résultats en dimension 2pour des écoulements monophasiques modélisés à l’aide de la loi des gaz raidis pour la phase liquide.Ces résultats sont obtenus grâce au code FreeFem++ basé sur la méthode des Éléments Finis.

1. Introduction

The modelling of nuclear reactors is hard to achieve since it requires the coupling of several multi-scale multi-physics problems [6]. Indeed, a reactor is characterized by a large number of systems corresponding to differentfunctions (heating, cooling, energy production, . . . ). The natural process is thus to split the problems intolower-scale ones and then to carry out the coupling between them [11]. Moreover, there exist several industrial

1 DEN/DANS/DM2S/STMF, Commissariat à l’Énergie Atomique et aux Énergies Alternatives – Saclay, 91191 Gif-sur-Yvette,France & e-mail: [email protected] Université de Toulon – IMATH, EA 2134, avenue de l’Université, 83957 La Garde, France& e-mail: [email protected] MAP5 UMR CNRS 8145 - Université Paris Descartes - Sorbonne Paris Cité, 45 rue des Saints Pères, 75270 Paris Cedex 6,France & e-mail: [email protected] & e-mail: [email protected] CEREMA-INRIA – team ANGE and LJLL UMR CNRS 7598, 4 place Jussieu, 75005 Paris, France& e-mail: [email protected]

© EDP Sciences, SMAI 2014

Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201445014

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ESAIM: PROCEEDINGS AND SURVEYS 139

codes based on a compressible model providing numerical approximations of the whole nuclear reactor (see forinstance [1, 4]).

A model was proposed in [7] to describe specifically flows within the reactor core in a simplified approach (whichmay be enriched to provide a more complete description of the overall process). Based on the assumption thatthe average Mach number is small, the model was derived through an asymptotic expansion performed in themonophasic compressible Navier-Stokes equations with an energy source term. The readers may refer to [12,15]for similar processes leading to low Mach number models. This asymptotic approach amounts to filtering outthe acoustic waves. Consequently, the mathematical nature of the resulting system of PDEs is modified whichrequires numerical methods that are different from the compressible Navier-Stokes framework. The modelderived in [7] is called Lmnc (for Low Mach Nuclear Core model) and consists of a transport equation upon athermodynamic variable (here the total enthalpy), of an elliptic divergence constraint upon the velocity (with asource term which underlines the compressible property of the flow) and of the parabolic momentum equation. Itthus exhibits a structure similar to the incompressible Navier-Stokes equations for which several Finite Elementalgorithms have been designed in the literature [16].

Applications to low Mach number configurations have been achieved for instance in [9]. Our study differs fromthe latter reference due to the model – which, in our case, is modified to match the low Mach regime (likein [5]) – and due to the boundary conditions imposed by the underlying nuclear framework. The structure ofthe equations lead to explicit analytic solutions [3] when the phases are modelled by the stiffened gas law evenwhen phase transition is involved. Notice that the present paper is restricted to monophasic flows as we aimat assessing a new numerical approach. 1D simulations were performed in [2, 3, 8] by means of a numericalscheme based on the method of characteristics. The latter algorithm strongly relies on the decoupling of theequations which is only valid in dimension 1. It could though be extended to dimension 2 through an explicittreatment. However, we did not select this strategy in the present work. We rather choose to apply the FiniteElement method directly by using the free software FreeFem++ [10]. This robust tool enables to deal withthe aformentioned boundary conditions. Contrary to [9], the convective part of the equations is not treateddirectly through the weak formulation but by means of the method of characteristics [14]. We emphasize thatno theoretical study of the Finite Element approach is carried out in this paper as we only focus on numericalsimulations.

The paper is organized as follows. The model is presented in Section 2 including the initial and boundaryconditions. Its weak formulation is given in Section 3 as well as the resulting algorithm. Academic numericalexamples are then provided in Section 4, namely an intrinsically 1D flow (which enables to legitimate thenumerical strategy) and a pure 2D flow.

2. Governing equations

The model is set in the rectangular domain Ω = x = (x, y) ∈ [0, Lx] × [0, Ly]. The 2D nonconservativeformulation of the Lmnc model [7] reads

∇ · u =β(h, p0)

p0Φ(t,x), (1a)

ρ(h, p0) · (∂th+ u · ∇h) = Φ(t,x), (1b)

ρ(h, p0) ·(∂tu + (u · ∇)u

)−∇ · σ(u) +∇p = ρ(h, p0)g, (1c)

where u = (u, v) and h denote respectively the velocity field and the total enthalpy of the fluid. This modelis characterized by two pressure fields. This decomposition results from the filtering out of the acoustic waves.The thermodynamic pressure p0 is involved in the equation of state and is an average pressure (constant in time

140 ESAIM: PROCEEDINGS AND SURVEYS

and space) within the core. The dynamic pressure p appears in the momentum equation and can be considereda perturbation around p0. We mention that model (1) is only valid under the assumption that pressure p0 doesnot depend on time.

The stress tensor σ(u) models viscous effects: the classic internal friction in the fluid as well as the friction onthe fluid due to technological devices in the nuclear core (e.g. the friction on the fluid due to the fuel rods). Inthe sequel, we take

σ(u) = µ(h, p0)(∇u + (∇u)T

)+ η(h, p0)(∇ · u) I,

where µ and η are the viscosity coefficients determined by constitutive laws. Other choices are possible dependingon the modelling scale at stake.

The power density Φ(t,x) is a given function of time and space modelling the heating of the coolant fluid dueto the fission reactions in the nuclear core. Finally, g is the gravity field.

To close the system, we have to specify the equation of state (EOS) that relates the density ρ to the unknownsof System (1). In the present work, the properties of the monophasic fluid are prescribed by the stiffened gaslaw:

ρ(h, p0) =γ`

γ` − 1

p0 + π`h− q`

(2)

where γl, q` and π` are characteristic constants of the liquid phase [3, Table 1].

This relation yields the expression of the compressibility coefficient involved in Equation (1a)

β(h, p0) def= − p0

ρ2(h, p0)· ∂ρ∂h

(h, p0) = β`(p0) def=γ` − 1

γ`

p0

p0 + π`. (3)

The coefficient is constant in the case of this equation of state.

Boundary and initial conditions

The fluid is injected at the bottom of the core (y = 0) at a given enthalpy he and at a given flow rate De.Without viscous effects, it is possible to impose the dynamic pressure p at the exit of the core (y = Ly) as in [3].When viscosity is taken into account, the nature of the equations is different and more information is neededat the exit. We chose here a free outflow boundary condition.

More precisely, the boundary conditions (BC) at the bottom of the domain areh(t, x, 0) = he(t, x),

(ρu)(t, x, 0) = (0, De(t, x)),(4)

and at the top of the domain we consider free outflow conditions

(σ(u)n− pn)(t, x, Ly) = 0, (5)

where n is the unit normal vector. On the lateral walls we consider free-slip condition, i.e.(u · n)(t, 0, y) = (u · n)(t, Lx, y) = 0, (6a)(σ(u)n · τ )(t, 0, y) = (σ(u)n · τ )(t, Lx, y) = 0, (6b)

where τ is some unit tangential vector.


As for the initial state, it is prescribed by h(0,x) = h0(x),

u(0,x) = u0(x),

p(0,x) = 0.

(7)

Hypotheses

Several assumptions are made to ensure that the problem is well-posed and has a physical meaning. As for thedata, we suppose that:

(i) Φ(t,x) is nonnegative for all (t,x) ∈ R+ × [0, Lx]× [0, Ly];(ii) p0 is a positive constant.

Hyp. (i) characterizes the fact that we study a nuclear core where the coolant fluid is heated.

The second hypothesis concerns the modelling parameters:

(iii) γ` > 1;(iv) π` is such that p0 + π` > 0;(v) µ and η are equal to two constants µ0 > 0 and η0 satisfying 2µ0 + 3η0 > 0.

Notice the previous assumptions ensure that β` > 0.

Finally, the initial/boundary states involved in (4-7) are constrained by:

(vi) De(t, x) > 0 for all t ≥ 0 and x ∈ [0, Lx];(vii) he(t, x) > q` for all t ≥ 0 and x ∈ [0, Lx];(viii) h0 is such that h0(x, y = 0) = he(t = 0, x) for all x ∈ [0, Lx] and h0(x) > q` for all x ∈ [0, Lx]× [0, Ly];(ix) u0 is such that ∇ · u0 =

β`(p0)

p0Φ(0,x),

u0(x, y = 0) = ue(t = 0, x),

and satisfies BC (5) and (6).

Assumption (vi) corresponds to a nuclear power plant of PWR or BWR type: the flow is upward1. Assump-tion (vii) means that EOS (2) is such that ρ(he, p0) is well-defined and positive. This enables to compute theinflow velocity ue by

ue(t, x) def= 0, ve(t, x) =De(t, x)

ρ(he(t, x), p0

) .Likewise, Hyp. (viii) leads to the existence of ρ(h0, p0) through (2). Finally, Hyp. (ix) corresponds to the factthat the steady equation (1a) is initially satisfied, which means that initial conditions are well-prepared (see [3]for instance).

1The flow could be downward when the nuclear reactor is a material testing reactor. The present model still applies but withadapted BC.


3. Weak formulation and numerical scheme

To provide a numerical approximation of the solutions to System (1), we carry out a standard Finite Elementapproach in order to use the FreeFem++ software [10]. As for the convective part, it can be treated byincorporating a trilinear form into the weak formulation as in [9]. However, in keeping with previous works [2,3]and as FreeFem++ incorporates this feature, we decided to apply the method of characteristics [14].

Let us define the characteristic flow X as the solution of the ordinary differential equation (ODE)dXdτ

(τ ; t,x) = u(τ,X (τ ; t,x)

),

X (t; t,x) = x,(8)

parametrized by t ∈ R and x ∈ Ω. A straightforward remark is that for any field ζ : R× Ω→ Rp(d

dτ

[ζ(τ,X (τ ; t,x)

)])|τ=t

= ∂tζ(t,x) +(u(t,x) · ∇

)ζ(t,x).

Hence, the convective part in Equations (1b) and (1c) can be approximated for any ∆t > 0 by

[∂t +

(u(t,x) · ∇

)](hu

)(t,x) ≈ 1

∆t

[(hu

)(t,x)−

(hu

)(t−∆t,X (t−∆t; t,x)

)].

The algorithm induced by this technique thus consists in finding (h,u, p)(t, ·) ∈ (he +H) × (ue + U) × L2(Ω)such that for all (θ,v, ψ) ∈ H × U × L2(Ω)

•∫

Ω

ψ∇ · u dx =β`p0

∫Ω

Φψ dx,

•∫

Ω

h(t,x)− h(t−∆t,X (t−∆t; t,x)

)∆t

θ(x) dx =

∫Ω

Φ(t,x)

ρ(h(t,x), p0

)θ(x) dx,

•∫

Ω

ρ(h(t,x), p0

)u(t,x)− u(t−∆t,X (t−∆t; t,x)

)∆t

· v(x) dx

+µ0

2

∫Ω

(∇u + (∇u)T

)::(∇v + (∇v)T

)dx+ η0

∫Ω

(∇ · u)(∇ · v) dx−∫

Ω

p∇ · v dx

=

∫Ω

ρ(h(t,x), p0

)g · v(x) dx,

where

H =θ ∈ H1(Ω): θ(x, 0) = 0

,

U =v ∈

(H1(Ω)

)2: v(x, 0) = 0, v · n(0, y) = v · n(Lx, y) = 0

.

Boundary integrals vanished either due to BC (5) and (6b) or to the functional spaces including homogeneousversions of (4) and (6a).

The theoretical investigation of the weak formulation is not the topic of this paper. However, we just specify thechoice of H. The first reason is to ensure the existence of the trace of h ∈ H on y = 0 in order to satisfy (4).The second one is that it enables to prove that h − q` > minh0 − q` through the weak formulation of the


(a) Normalized error upon the density –Scale ranges from 1.57× 10−6 (orange)to 5.83× 10−5 (red)

(b) Convergence curves: L2 errors with respect to the meshsize

Figure 1. Comparisons with exact solutions. Colors range from orange to yellow, to green,to blue and then to red.

transport equation and thus that ρ(h, p0) is positive and belongs to L∞(Ω). The latter point shows that theintegral

∫Ωρ(h, p0)u · v dx is well-defined. We recall that ρ(h, p0) is defined by EOS (2).

As u is an unknown of the overall problem, the exact solution of ODE (8) cannot be achieved. There existseveral numerical techniques to yield an approximation of X (t − ∆t; t,x) (see [13, 14]). In the sequel, thisapproximation will be denoted by ξ no matter what method is used to compute it. The corresponding routinein FreeFem++ is convect.

Given a time sampling t0, t1, . . ., tn, we consider the following semi-implicit discretization: find(hn+1,un+1, pn+1

)∈

(he +H)× (ue + U)× L2(Ω) such that for all (θ,v, ψ) ∈ H × U × L2(Ω)

•∫

Ω

∇ · un+1ψ dx =β`p0

∫Ω

Φ(tn,x)ψ dx, •∫

Ω

hn+1 − hn(ξn)

∆tθ dx =

∫Ω

Φ(tn,x)

ρ(hn, p0)θ dx,

•∫

Ω

ρ(hn, p0)un+1 − un(ξn)

∆t· v dx+

µ0

2

∫Ω

(∇un+1 + (∇un+1)

T)

::(∇v + (∇v)T

)dx

+ η0

∫Ω

(∇ · un+1)(∇ · v) dx−∫

Ω

pn+1∇ · v dx =

∫Ω

ρ(hn, p0)g · v dx.

This weak formulation is discretized in space on a triangular mesh. It is then solved with the FE–softwareFreeFem++. We do not give more details about this step in the present paper. We only mention that thetime step must be computed as a function of the mesh size in order to ensure convergence.2


4. Numerical examples

Numerical simulations of 1D single-phase flows are provided in [2]. Here, we focus on 2D flows and perform somesimulations obtained by means of FreeFem++ (using P1 Lagrange finite elements) applied to the discrete weakformulation presented in Section 3. These results are aimed at providing exploratory hints about the behaviourof solutions to our Lmnc model in dimension 2.

Two data sets are considered: the first academic test enables to assess the numerical approach insofar asan analytic monodimensional solution must be recovered. The second test presents real 2D effects due to anonvertical gravity field. Parameters are set as follows:

• Geometry of the domain: Lx = Ly = 1 m.• Parameters involved in EOS (2) for the pure liquid: γ` = 2.35, π` = 109 Pa, q` = −1167.056× 103 J · kg.• Reference value for pressure, gravity intensity and power density: p0 = 155 × 105 Pa, g = 9.81 m · s−2,

Φ0 = 170× 106 W ·m−3.• Inflow data: he = 1.236508× 106 J ·K−1, ve = 5 m · s−1.

4.1. Recovering 1D solutions

In this first case, the setting is such that the analytic 1D solution derived in [3, Sect. 3.2] is also the solution ofthe 2D problem. It corresponds to the following choice for the parameters:

• Power density: Φ(t,x) = Φ0.• Gravity field: g = g × (0,−1).

The result displayed on Figure 1(a) shows the pointwise error over the domain between the analytic solutionρ(h∞(y), p0

)from [3, Remark 3.1] duplicated for each x ∈ [0, Lx] and the numerical solution when the asymptotic

state is reached (here t = 0.3 s). The corresponding mesh is made of 952 triangles and 517 nodes while the meanerror is about 10−6. The error is not uniform due to the unstructured mesh. We also performed a refinementprocess which leads to an order 1 convergence towards the monodimensional analytic solution – see Fig. 1(b).

The numerical strategy using FreeFem++ and detailed above thus enables to recover the expected solutionwhich legitimates the approach.

4.2. Genuine 2D flows

We now focus on a test which displays 2D phenomena. To do so, we modify the data:

• Power density: Φ(t,x) = Φ0 × 104 × exp(

−1r20−|x−x0|2

)1|x−x0|<r0, x0 = ( 1

2 ,12 ) and r0 = 0.4 – see

Fig. 2(a).• Gravity field: g = g

√2

2 × (1,−1).

This corresponds to a localized heating at the center of the core. We notice on Figs. 2(c) and (d) the effectsof the convection compared to Figs. 3(a) and (b). The rise of temperature is not restricted to the supportof Φ and a steady state can be reached. The influence of the nonvertical gravity field is also noticeable onFigs. 2(b) and (g). Indeed, we observe on Fig. 2(b) that the pressure increases at the low right part of the corewhich induces a dissymmetry on Fig. 2(g). Fig. 2(f) also shows the x-component of the velocity is no longer

2The time step is set as follows: ∆t = C δ|ue|

where δ is a characteristic mesh size and C ∼ 1 is a constant.


(a) Mesh and density power– Scale ranges from0 W ·m−3 (orange) to3.38× 109 W ·m−3 (red).

(b) p−minΩ

p – Scale ranges from 0 Pa (orange) to 8459 Pa (red).

(c) Temperature – Scale ranges from 561 K (orange) to621K (red).

(d) Density – Scale ranges from 667 kg ·m−3

(orange) to 738 kg ·m−3 (red).

(e) Mach number – Scaleranges from 2.33×10−3 (or-ange) to 3.3× 10−3 (red).

(f) Velocity: x-component– Scale ranges from0.014 m · s−1 (orange) to0.55 m · s−1 (red).

(g) Velocity: y-component– Scale ranges from4.25 m · s−1 (orange) to5.92 m · s−1 (red).

Figure 2. Simulations of a confined heating. Colors range from orange to yellow, to green, toblue and then to red.

negligible. The fact still remains this experiment matches the low Mach number assumption as it is highlightedon Fig. 2(e).


(a) Temperature – Scale ranges from 556 K (orange) to806 K (red).

(b) Density – Scale ranges from 512 kg ·m−3

(orange) to 742 kg ·m−3 (red).

Figure 3. Test case: model without convective terms. Colors range from orange to yellow, togreen, to blue and then to red.

5. Conclusion & Perspectives

The 2D numerical results presented in this paper show that the Finite Element method coupled to a methodof characteristics is suitable to solve a low Mach system which models the heat transfer in a nuclear core.It is achieved although the variety of nature of the equations involved in the system. The specific boundaryconditions are correctly taken into account through the weak formulation.

This work is a first step in simulating the Lmnc model after some 1D devoted papers [2, 3, 7]. Other stepsmust follow on, including the proof of numerical properties such as positivity preservation, convergence resultsand error estimates. Moreover, the modelling of the nuclear core will have to be improved, in particular byintroducing phase transition using tabulated equations of state, which will be achieved in [8].

References

[1] TRACE V5.0 Theory Manual, Field Equations, Solution Methods and Physical Models. Technical report, U.S. Nuclear Regu-latory Commission, 2008.

[2] M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec, O. Lafitte, T.-T. Nguyen, and Y. Penel. Study of low Mach nuclear coremodel for single-phase flow. ESAIM Proc., 38:118–134, 2012.

[3] M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec, and Y. Penel. Study of low Mach nuclear core model for two-phase flowswith phase transition I: stiffened gas law. Math. Model. Numer. Anal., 48(6):1639–1679, 2014.

[4] D. Bestion. The physical closure laws in the CATHARE code. Nucl. Eng. Des., 124(3):229–245, 1990.[5] M. Braack and R. Rannacher. Adaptive finite element methods for low Mach number flows with chemical reactions. IWR,

1999.[6] J.M. Delhaye. Thermohydraulique des réacteurs. EDP sciences, 2008.[7] S. Dellacherie. On a low Mach nuclear core model. ESAIM Proc., 35:79–106, 2012.[8] S. Dellacherie, G. Faccanoni, B. Grec, and Y. Penel. Study of low Mach nuclear core model for two-phase flows with phase

transition II: tabulated EOS. In preparation.[9] T. Fortin. Une méthode d’éléments finis à décomposition L2 d’ordre élevé motivée par la simulation des écoulements diphasiques

bas Mach. PhD thesis, Univ. Paris 6, 2006.[10] F. Hecht. FreeFem++, 3.25 edition, 2013.[11] Olivier Hurisse. Couplage interfacial instationnaire de modèles diphasiques. PhD thesis, Univ. Aix-Marseille I, 2006.[12] A. Majda and J. Sethian. The derivation and numerical solution of the equations for zero Mach number combustion. Combust.

Sci. Technol., 42(3-4):185–205, 1985.[13] Y. Penel. An explicit stable numerical scheme for the 1D transport equation. Discrete Contin. Dyn. Syst. Ser. S, 5(3):641–656,

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[14] O. Pironneau. On the transport-diffusion algorithm and its applications to the navier-stokes equations. Numer. Math., 38:309–332, 1982.

[15] G.I. Sivashinsky. Hydrodynamic theory of flame propagation in an enclosed volume. Acta Astronaut., 6:631–645, 1979.[16] C. Taylor and P. Hood. Numerical solution of the Navier-Stokes equations using the finite element technique. Comput. &

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Stéphane Dellacherie , Gloria Faccanoni , Bérénice Grec ... · j x x 0j2 1 fj x 0j